Teaching
& learning
Additional mathematics Form 4
NAME:…………………………………………………. FORM :…………………………………………………
Date of gatting the module: …………………………………….. Date of completing the module …………………………. Marks of the Topical Test : ……………………………..
Prepared by : Addational Mathematics Department Sek Men Sains Muzaffar Syah Melaka For Internal Circulations Only
In this subtopic subtopic you will will learn to : 1. Understand the concept of quadratic quadratic equation and and its roots 1.1 Recognise a quadratic quadratic equation and express express it in general general form. 1.2 Determine whether whether a given given value is the root of a quadratic equation equation by a) substitution; b) inspection. 1.3 Determine roots roots of quadratic quadratic equations equations by trial and and improvement improvement method .
1. Understand the concept concept of quadratic equation and its roots. roots. 2 Note : The general form of a quadratic equation is ax + b x + c = o. where a ,b, and c are constants and a
≠
0 . The highest power of the unknown, ( x ) , is 2
1. State whether each of the following equations is a quadratic equation or not.
Equations 1. 2x + 4 = 0
yes or no no
Give your reason The highest power of x is one .It’s a linear equation
2. x 2 + 4 = 0 3. 3x 2 + 2x + 3 = 0 4. 5x = 2 - 3x 5. y (2 (2 – 3y ) = 7 6. p( 3p – 2) = 4 + 2p 7.
1
– 2x = 0 x 2 8. (x – 7)2 = 6 9. 5xy + 6 = 0 10. x 3 + 2x = 4 11. ( n + 2)(n – 3) = 5 12.
2 x
+
4x2
−3 =
0
1.1 Recognise a quadratic equation equation and express express it in general form. 1.1 Rewrite the following following quadratic equations in general general form and and find find the value of a, b and c . Example 1 x – 2x = 3 2
x 2 – 2x – 3 = 0
Example 2 (3x + 1)(x -3) -3) = 4
Example 3
x 2
+1 =
10 x 3
a = 1, b = – 2, c = – 3
2
a)
2x 2 = 3x – 4
a)
(2x – – 1) = x + 5 x (2
b) (3 x + 2) = 8
c) x 2
c)
d) 3 x
5 x − 3 p
2
+
2
= x + 5k − 3
e) 4 x
4 x
= 1−
2 p
2 3
( x 2
− 3 x) = x
2
b) n (2 (2n -1 ) = 3n
−1 =
a)
d) x
3 x 2 – 5 = 4x (1 (1 –x )
2
e) 5 x
2
+
2mx
− x =
=
3x − 5
3kx − 4
b) 2 x 2
c) 2 x 2
d) 2 x
−1 =
+
x
3
=
1
x 4
5
2
+ 5 x = p (1 −
2
+
e) px
4 x + 3q
2 x)
= 1+
2 x
Homework : Text Text Book page 26 Exercise 2. 1.1
1.2 Determine whether whether a given value is the root of a quadratic quadratic equation by a) substitution; b) inspection. Note : The The root of a quadratic equation is the value value o the unknown unknown in the equation which satisfies satisfies the equation . If a value is given, it can be determined whether it is a root by substitution or inspection.
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1 Determine Determine whether the x value given given are the roots of the following following quadratic equations equations .
Quadratic equations
x value 12 – 2(1) – 3 = –4
x = 1,
(–1)2 – 2(–1) – 3 = 0
x = –1,
Root/ No No Yes
1. x 2 – 2x – 3 = 0
3 2 – 2(3) – 3 = 0
x = 3,
(–3)2 – 2(–3) – 3 = 12
x = –3,
x = 1,
x = 3, 2. 3x 2 – 5x – 12 = 0 x = –3,
x=–
4 3
,
x = 1,
x = 4, 3. (2 x + 1 ) ( x- 4 ) = 0 x = –3,
x=–
1 2
,
Homework : Text Book page 28 Exercise 2. 1.2 1.3 Determine roots of quadratic quadratic equations equations by trial and improvement improvement method. Trial and and improvement improvement method is a primitive primitive method of repeated repeated substitution substitution of integers integers into a function or polynomials to find solutions. (Synonymous to trial and error method) Example 1.3 Find the roots of the quadratic equation x2 - 5x + 6 = 0 by using trial and improvement method metho d Solution: Trial x x2 - 5x + 6 Trial x x2 - 5x + 6 First -3 Fourth 2 Second 3 Fifth -2 Third 1 Sixth 6 Homework : Text Book page 29 Exercise 2. 1.3 and Skill Practice 2.1 page 29
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What we should learnt in this subtopic are : 2. Understand the concept of quadratic equations
.
2.1 Determine the roots of a quadratic equation by a) factorisation; b) completing the square c) using the formula. 2.2 Form a quadratic equation from given roots
2.1 Determine the roots of a quadratic equation by a) factorisation; b) completing the square c) factorisation factorised ax 2 + b x + c = o can be factorised completely by converting converting on on the left hand side as a product of two linear factor.
( x − a )( x − b ) = 0 x − a x
=
a
=
0 or x − b or x
=
=
b
0
using the formula d) using calculator
completing the square completing the square is Converting an expression or equation into the "perfect square" form. Converting from general general form to perfect square form 2 i.e. y = ax + bx + c to 2 y = (x + a ) + d 2
Using a formula Quadratic Equations can also be solved by using the formula as follows :
x
=
−b±
b2
−
4ac
2a
where
a ,b, c are constants
2
Eg. 2x - 8x +5 +5 = 2(x -2) -2) - 3
It used to find the roots of quadratic equations when the quadratic equations , cannot factorise.
and related to ax 2 + bx + c =0
Example 2.1 a) Solve the equation
(2 x + 1) (3x − 2) = 3
2
b) Solve the equation x by completing the square
+ 5x −
2=0
.
by factorisation .
d) Solve Solve the equation equation
4 x
2
+ 12 x + 9 =
factorisation
0 by
2
Solve the equation 2 x − 4 x − 3 = 0 by completing the square .
c) Solve the equation
x 2 − 10 x − 3 = 0 by using a formula .
Solve the equation
3 x( x − 2) − x( x − 11) = 3 by using a formula .
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Exercise 2.1 1) Solve the following quadratic equation by factorisation. factorisation.
a) . 12 x 2
Ans:
−5
4
+
,
b) . 9 y 2
7 x = 10
= 12 y −
4
Ans:
2
2 3
3
answers correct 2. Solve the following quadratic equation by completing completing the square. Give your answers to 4 significant figure
a) . 2 y 2
+
b) . ( x − 1) ( x − 5) = 25
6y + 3 = 0
Ans: -2.385; 8.385
Ans : -2.366; -0.634
answers correct to 4 3. Solve the following quadratic equation by using formula . Give your answers significant figure
A ) 5 x 2
=
b) 5 x 2
2x + 2
=
2x + 2
Ans:-0.463;0.863
1 Ans: ,-3 2
Note : Solve all quadratic equations above using calculator
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Homework : Text Book page 32 Exercise 2. 2.1 2.2 Form a quadratic equation from given given roots 1. If a and b are the roots of a quadratic equation then x = a or x = b x – a = 0 or x – b = 0 (x – a )( )(x – b ) = 0, hence x 2 – (a + b )x )x + ab = 0 Therefore , the quadratic equation with roots P and q is x 2 – ( a+b) x + ab = 0 = 0 2. The Step of forming a quadratic equation from given roots are i. ii. iii.
Find the sum of the roots Find the product of the roots Form a quadratic equation by writing in a following form 2
x
– ( sum of the roots )
+ product of the roots = 0
x
Example 2.2(i) Form the quadratic equation equation whose roots are shown below below
a).
3 and -5
Exercise 2.2 1 a) . − 7
b).
b).
4 and
1 2
2
c) 3 α and 2α
3
and − 2
1
c) 4r and 5r
3
Example 2.2(ii) State the sum and product of the roots of the following quadratic equations. equations. 2 2 a) . x - 9x - 4 = 0 b). 3x + 5x + 4 = 0 – 1) = 2(1 – x ) c) x (x –
Exercise 2.2(ii) State the sum and product of the roots of the following following quadratic equations. equations. 2 2 2 b). 2x - 6x + 3 = 0 c) ). 2x +( t +2) x + t 2 = 0 a) . x + 4 x + 5 = 0
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Example 2.2(ii) If α and β are the roots of the equation 2x 2 + 5x – 6 = 0, Form the equations whose roots are a) α + 1 , β + 1
b)
2
,
2
α β
Exercise 2.2(ii) If α and β are the roots of the equation 3x 2 - 2x + 4 = 0, Form the equations equations whose whose roots are a) 2α + 1 , 2β + 1
b)
1
,
1
α β
Example 2.2(iii) 1. If One root of the equations 27x2 + kx – 8 = 0 is square the other .Find the value of k .
Exercise 2.2(iii) 2 2. If one root of the equations 2 x + x – c = 0 is two times the other, find the value of c
8
3. Find q if the equation 3x 2 - 4x + q = 0 has equal equal roots
2
4. If the roots of the equation equation x2 + px + 7 = 0 are denoted by α and β , and α .find the possible values of p (camb)
+
β 2
=
22
5. Given that α and β are the roots of the equation equation x2 – 2x + 3 = 0, Find a quadratic equation equation 2
whose roots are α
2
+ β
and 2αβ (Camb)
6. . Given that α and β are the roots of the equation equation 2x2 – 3x + 4 = 0,Write down the value of
α + β and 2αβ . Find an equation whose roots are α +
1 α
and β +
1 β
(camb)
Homework : Text Book page 34 Exercise 2. 2.2 and Skill Practice 2.2
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3. Understand and use the conditions for quadratic equations to have a) two different roots; b) two equal roots; c)
no roots
/ no real roots
3.1 Determine types of roots of quadratic equations from from the value of b2 − 4ac. 3.2 Solve problems involving 2 b − 4ac in quadratic equations to: a) find an unknown value; b) derive a relation.
3.1 Determine types of roots of quadratic equations from from the value of b2 − 4ac. For the quadratic equation ax 2 + b x + c = o, the discriminant of the equation is Types of roots of quadratic equations from the value of b 2 4ac 2 (i) b 4ac > 0 ….Two different roots ( the roots are distinct) (ii) b 2 4ac = 0 …Two same roots 2 (iii) b 4ac < 0 …. No real roots
2
b
4 ac
example 3.1 Determine the type of the roots of the following following quadratic equations equations 2 2 ax + bx + c = 0 a b c b -4ac Type of roots Two different roots 1 5 6 1 2 1. x + 5x + 6 = 0 2. x 2 + 6x + 9 = 0 3. 4x 2 - 4x + 1 = 0 4. 2x 2 - 4x - 5 = 0 5. 2x 2 - 5x + 4 = 0 Homework : Text Book page 36 Exercise 2. 3.1. 3.2 Solve problems involving b2 − 4ac in quadratic equations to: a) find an unknown value; b) derive a relation. The value b2 − 4ac can be used to find the unknown value of coefficients or to derive a relation which involves unknown in the quadratic equations
Example 3.2 a) Find k if x 2 + 8x + k = 0 has equal roots
b) Find p if 3x 2 + 2x + 3p = 0,has two different roots
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Exercise 3.2
a) Find The range of value of p if 3x 2 – 1 = 6x – 2p has two distinct roots
Find The range of value of p if has two different roots x 2 + 2x +9 = p(2x – p) has
[p<2] c) .The quadratic equation 3x + 2x + x + h = 0 has equal roots roots . Find the value of h 2
[h=
1 3
d) Show that the roots of the equations 6x – 6 -2px 2 = x 2 are complex if p >
1 4
]
e) Given that x 2 +(p - 2)x + x + 10 - p = 0 has two equal roots, find the values of p .
[P=
[P > - 4]
Find the range of values of h for the quadratic equation 2x 2 + 3x +4p = 1 which has no roots
±
6 ]
[p>
17 32
]
Homework : Text Book page 37 Exercise 2. 3.2. and skill Practice 2.3
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SPM QUESTIONS
SPM 2003 [ 2.591 or - 0.2573 ] Solve the quadratic equation 2 x( x − 4) = (1 − x)( x + 2) . Give your answer correct to four significant significant figure figure [ 2 marks marks ]
SPM 2004 Form the quadratic equations which has the roots – 3 and
1
. Give your answer in the form ax 2 + bx + c
2 = 0 , where a,b and c are constants
[ 2 marks ]
SPM 2004 Form the quadratic equations which has the roots – 3 and
1
. Give your answer in the form ax 2 + bx + c
SPM 2003 [ p < -3 or p > 5] The quadratic equation x( x + 1) = px − 4 has two distinct roots. Find the range of values of p
2 = 0 , where a,b and c are constants
SPM 2005 Solve the quadratic equation x(2 x − 5) = 2 x − 1 Give your answer correct to three decimal place
SPM 2006 2 A quadratic equation x + px + 9 = 2 x has two equal roots. Find the possible values of p
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